Differential k-form - Knowlpedia

Let $M$ be a smooth manifold and let $\pi:T^*M\to M$ be its cotangent bundle.

Definition. A differential $k$ -form on $M$ is a smooth section of the exterior power bundle $\Lambda^k T^*M\to M$ . For $k\ge1$ , concretely it is a rule that assigns to each $p\in M$ an alternating $k$ -linear map

\omega_p:(T_pM)^k\to \mathbb{R},

depending smoothly on $p$ (here $T_pM$ is the tangent space).

The set of all smooth $k$ -forms is denoted $\Omega^k(M)$ . For $k=0$ , one has $\Omega^0(M)=C^\infty(M)$ . For $k=1$ , a $1$ -form is the same thing as a smooth covector field (a smooth section of $T^*M$ ), and its behavior under smooth maps is governed by the pullback of covectors (more generally by the pullback of differential forms).

Local expression

In a smooth chart $(U,x^1,\dots,x^n)$ , every $k$ -form can be written uniquely as

\omega = \sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\cdots i_k}\, dx^{i_1}\wedge\cdots\wedge dx^{i_k} \quad\text{on }U,

with smooth coefficient functions $a_{i_1\cdots i_k}\in C^\infty(U)$ , using the wedge product.

Two fundamental operations on forms are the wedge product $\wedge$ and the exterior derivative $d:\Omega^k(M)\to \Omega^{k+1}(M)$ , which leads to the notions of closed forms, exact forms, and de Rham cohomology.

Examples

A $0$ -form and its differential. Any smooth function $f\in C^\infty(M)$ is a $0$ -form. Its exterior derivative $df$ is a $1$ -form characterized by $df_p(v)=v(f)$ for $v\in T_pM$ .

A $1$ -form on $\mathbb{R}^2$ . On $\mathbb{R}^2$ with coordinates $(x,y)$ , the expression $\omega = x\,dy - y\,dx$ defines a smooth $1$ -form. At each point $(x,y)$ , it is a covector that eats a tangent vector $(u,v)$ and returns $xv-yu$ .

Standard volume form on $\mathbb{R}^n$ . On $\mathbb{R}^n$ with coordinates $x^1,\dots,x^n$ , the $n$ -form $dx^1\wedge\cdots\wedge dx^n$ is a nowhere-vanishing differential form (a smooth choice of oriented volume density).